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Graph Theory Open

Seymour's Second Neighborhood Conjecture

In every oriented graph, is there a vertex whose out-neighborhood's second out-neighborhood is at least as large as its first? A clean statement about local expansion in orientations.

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  1. Validated increment

    Settled for tournaments; open for general orientations

    The conjecture is known for tournaments and several sparse or locally restricted oriented classes. The general oriented-graph case remains the target.

    Previous best

    Open even for tournaments (historical)

    New certified

    Proved for tournaments; open in general

    Public literature baseline

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